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On the Continuity Properties of the Set of Trajectories of the Control System with Limited Control Resources

Year 2023, Volume: 19 Issue: 2, 151 - 157, 29.06.2023

Anar Huseyin

Abstract

In this paper the control system with integral constraint on the control functions is studied where the behavior of the system by the Urysohn type integral equation is described. The admissible control functions are chosen from the closed ball of the space $L_p ([a,b];R^m )$ $(p>1)$ centered at the origin with radius $r$. Dependence of the set of trajectories on r and p is investigated. It is proved that the set of trajectories is Lipschitz continuous with respect to r and continuous with respect to $p$. The robustness of the trajectory with respect to the fast consumption of the remaining control resource is established.

Keywords

control system Hausdorff distance integral constraint robustness Urysohn integral equation.

References

  • Aubin, J-P, Frankowska, H. Set Valued Analysis. Birkhauser: Boston, USA, 1990, pp 461.
  • Brauer, F. 1975. On a nonlinear integral equation for population growth problems. SIAM J. Math. Anal.; 69: 312-317.
  • Conti, R. Problemi di Controllo e di Controllo Ottimale. UTET: Torino, Italy, 1974, pp 239.
  • Deimling, K. Multivalued Differential Equations. Walter de Gruyter: Berlin, Germany, 1992, pp 260.
  • Guseinov, KG, Nazlipinar AS. 2007. On the continuity property of L_p balls and an application. J. Math. Anal. Appl.; 335: 1347-1359.
  • Gusev, MI, Zykov, IV. 2017. On extremal properties of the boundary points of reachable sets for control systems with integral constraints. Tr. Inst. Math. Mekh. UrO RAN; 23: 103-115.
  • Heisenberg, W. Physics and Philosophy. The Revolution in Modern Science. George Allen & Unwin: London, Great Britain, 1958, pp 176.
  • Huseyin, A. 2017. On the existence of ε-optimal trajectories of the control systems with constrained control resources. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.; 66: 75-84.
  • Huseyin, N, Guseinov, KG, Ushakov, VN. 2015. Approximate construction of the set of trajectories of the control system described by a Volterra integral equation. Math. Nachr.; 288(16): 1891-1899, 2015.
  • Huseyin, N, Huseyin, A, Guseinov KG. 2018 Approxmation of the set of trajectories of the nonlinear control system with limited control resources. Math. Model. Anal.; 23(1): 152-166.
  • Huseyin, N. 2020. On the properties of the set of p-integrable trajectories of the control system with limited control resources. Internat. J. Control; 93(8): 1810-1816.
  • Ibragimov, G, Alias, IA, Waziri, U, Jafaaru, AB. 2019. Differential game of optimal pursuit for an infinite system of differential equations. Bull. Malaysian Math. Sci. Soc.; 42(1): 391-403.
  • Kalman, RE. (1963) Mathematical description of linear dynamical systems. J. SIAM Control Ser. A; 1: 152-192.
  • 14 Kelley, JL. General Topology. Springer: New York, USA, 1975, 298.
  • Krasovskii, NN. Theory of Control of Motion. Linear Systems. Nauka: Moscow, USSR, 1968, pp 475.
  • Krasovskii, NN, Subbotin AI. Game-Theoretical Control Problems. Springer-Verlag: New York, USA,1988, pp 517.
  • Krasnoselskii, MA, Krein, SG. 1955. On the principle of averaging in nonlinear mechanics. Uspekhi Mat. Nauk; 10: 147-153.
  • Kostousova, EK. 2020 On the polyhedral estimation of reachable sets in the "extended'' space for multistage systems with uncertain matrices and integral constraints. Tr. Inst. Mat. Mekh; 26(1), 141-155.
  • Polyanin, AD, Manzhirov, AV. Handbook of Integral Equation. CRC Press: Boca Raton, FL, USA, 1998, pp 1108.
  • Pontryagin, LS, Boltyanskii, VG, Gamkrelidze, RV, Mishchenko, EF. The Mathematical Theory of Optimal Processes. John Wiley & Sons: New York, USA, 1962, 360.
  • Subbotin, AI, Ushakov, VN. 1975. Alternative for an encounter-evasion differential game with integral constraints on the players' controls. J. Appl. Math. Mech.; 39(3): 367-375.
  • Subbotina, NN, Subbotin, AI. 1975. Alternative for the encounter-evasion differential game with constraints on the momenta of the players controls. J. Appl. Math. Mech.; 39(3): 376-385.
  • Ukhobotov, VI, Izmestev, IV. 2018. Impulse differential game with a mixed constraint on the choice of the control of the first player. Tr. Inst. Math. Mekh. UrO RAN; 24(1): 209-222.
  • Urysohn, PS. 1923. On a type of nonlinear integral equation. Mat. Sb; 31: 236-255.

Year 2023, Volume: 19 Issue: 2, 151 - 157, 29.06.2023

Anar Huseyin

Abstract

References

  • Aubin, J-P, Frankowska, H. Set Valued Analysis. Birkhauser: Boston, USA, 1990, pp 461.
  • Brauer, F. 1975. On a nonlinear integral equation for population growth problems. SIAM J. Math. Anal.; 69: 312-317.
  • Conti, R. Problemi di Controllo e di Controllo Ottimale. UTET: Torino, Italy, 1974, pp 239.
  • Deimling, K. Multivalued Differential Equations. Walter de Gruyter: Berlin, Germany, 1992, pp 260.
  • Guseinov, KG, Nazlipinar AS. 2007. On the continuity property of L_p balls and an application. J. Math. Anal. Appl.; 335: 1347-1359.
  • Gusev, MI, Zykov, IV. 2017. On extremal properties of the boundary points of reachable sets for control systems with integral constraints. Tr. Inst. Math. Mekh. UrO RAN; 23: 103-115.
  • Heisenberg, W. Physics and Philosophy. The Revolution in Modern Science. George Allen & Unwin: London, Great Britain, 1958, pp 176.
  • Huseyin, A. 2017. On the existence of ε-optimal trajectories of the control systems with constrained control resources. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.; 66: 75-84.
  • Huseyin, N, Guseinov, KG, Ushakov, VN. 2015. Approximate construction of the set of trajectories of the control system described by a Volterra integral equation. Math. Nachr.; 288(16): 1891-1899, 2015.
  • Huseyin, N, Huseyin, A, Guseinov KG. 2018 Approxmation of the set of trajectories of the nonlinear control system with limited control resources. Math. Model. Anal.; 23(1): 152-166.
  • Huseyin, N. 2020. On the properties of the set of p-integrable trajectories of the control system with limited control resources. Internat. J. Control; 93(8): 1810-1816.
  • Ibragimov, G, Alias, IA, Waziri, U, Jafaaru, AB. 2019. Differential game of optimal pursuit for an infinite system of differential equations. Bull. Malaysian Math. Sci. Soc.; 42(1): 391-403.
  • Kalman, RE. (1963) Mathematical description of linear dynamical systems. J. SIAM Control Ser. A; 1: 152-192.
  • 14 Kelley, JL. General Topology. Springer: New York, USA, 1975, 298.
  • Krasovskii, NN. Theory of Control of Motion. Linear Systems. Nauka: Moscow, USSR, 1968, pp 475.
  • Krasovskii, NN, Subbotin AI. Game-Theoretical Control Problems. Springer-Verlag: New York, USA,1988, pp 517.
  • Krasnoselskii, MA, Krein, SG. 1955. On the principle of averaging in nonlinear mechanics. Uspekhi Mat. Nauk; 10: 147-153.
  • Kostousova, EK. 2020 On the polyhedral estimation of reachable sets in the "extended'' space for multistage systems with uncertain matrices and integral constraints. Tr. Inst. Mat. Mekh; 26(1), 141-155.
  • Polyanin, AD, Manzhirov, AV. Handbook of Integral Equation. CRC Press: Boca Raton, FL, USA, 1998, pp 1108.
  • Pontryagin, LS, Boltyanskii, VG, Gamkrelidze, RV, Mishchenko, EF. The Mathematical Theory of Optimal Processes. John Wiley & Sons: New York, USA, 1962, 360.
  • Subbotin, AI, Ushakov, VN. 1975. Alternative for an encounter-evasion differential game with integral constraints on the players' controls. J. Appl. Math. Mech.; 39(3): 367-375.
  • Subbotina, NN, Subbotin, AI. 1975. Alternative for the encounter-evasion differential game with constraints on the momenta of the players controls. J. Appl. Math. Mech.; 39(3): 376-385.
  • Ukhobotov, VI, Izmestev, IV. 2018. Impulse differential game with a mixed constraint on the choice of the control of the first player. Tr. Inst. Math. Mekh. UrO RAN; 24(1): 209-222.
  • Urysohn, PS. 1923. On a type of nonlinear integral equation. Mat. Sb; 31: 236-255.
There are 24 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Anar Huseyin 0000-0002-3911-2304

Publication Date June 29, 2023
Published in Issue Year 2023 Volume: 19 Issue: 2

Cite

APA Huseyin, A. (2023). On the Continuity Properties of the Set of Trajectories of the Control System with Limited Control Resources. Celal Bayar Üniversitesi Fen Bilimleri Dergisi, 19(2), 151-157.
AMA Huseyin A. On the Continuity Properties of the Set of Trajectories of the Control System with Limited Control Resources. CBUJOS. June 2023;19(2):151-157.
Chicago Huseyin, Anar. “On the Continuity Properties of the Set of Trajectories of the Control System With Limited Control Resources”. Celal Bayar Üniversitesi Fen Bilimleri Dergisi 19, no. 2 (June 2023): 151-57.
EndNote Huseyin A (June 1, 2023) On the Continuity Properties of the Set of Trajectories of the Control System with Limited Control Resources. Celal Bayar Üniversitesi Fen Bilimleri Dergisi 19 2 151–157.
IEEE A. Huseyin, “On the Continuity Properties of the Set of Trajectories of the Control System with Limited Control Resources”, CBUJOS, vol. 19, no. 2, pp. 151–157, 2023.
ISNAD Huseyin, Anar. “On the Continuity Properties of the Set of Trajectories of the Control System With Limited Control Resources”. Celal Bayar Üniversitesi Fen Bilimleri Dergisi 19/2 (June 2023), 151-157.
JAMA Huseyin A. On the Continuity Properties of the Set of Trajectories of the Control System with Limited Control Resources. CBUJOS. 2023;19:151–157.
MLA Huseyin, Anar. “On the Continuity Properties of the Set of Trajectories of the Control System With Limited Control Resources”. Celal Bayar Üniversitesi Fen Bilimleri Dergisi, vol. 19, no. 2, 2023, pp. 151-7.
Vancouver Huseyin A. On the Continuity Properties of the Set of Trajectories of the Control System with Limited Control Resources. CBUJOS. 2023;19(2):151-7.
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